Evaluating $\lim\limits_{x→t}\frac1{x-t}{(\frac{\mathrm d}{\mathrm dt}(\mathrm e^x)-e^t)}$

Question

$t=\lim_{x→t}\frac{\left(\frac{d}{dt}\left(e^x\right)-e^t\right)}{x-t}$ , find t
how can we get the solution/s for the equation? Also for what value of t would there be no solution? I used the chain rule to simplify $\frac{d}{dt}\left(e^x\right)$ by
$\frac{d}{dt}\left(e^x\right)$=$\frac{d}{dt}\left(e^x\right)\left(\frac{dx}{dx}\right)$=$\frac{e^xdx}{dt}$
Then I tried using the L-hopital's rule, but couldnt reach anywhere.....
For the next part of the question I thought that the equation may not have a solution for t=e, while plotting the graphs, which turned out to differ in the answer and (and I think also it will in the number of solutions, as the) for varying values of t, still unable to understand how to find out the answer in terms of t.
I tried to plot the following on graph, at t=e which turned out to be--
For other values of t it turned out to be---
Please help me to find out the solution.

Answer 1

$x$ is independent of $t$ and $\dfrac{d}{dt}e^x=0$. The limit does not exist.